Problem 94 - Applied Calculus

Question

The rate of change of the volume of blood in the aorta \(t\) seconds after the beginning of the cardiac cycle is \(-k P_{0} e^{-m t}\) milliliters per second, where \(k, P_{0},\) and \(m\) are constants (depending, respectively, on the elasticity of the aorta, the initial aortic pressure, and various characteristics of the cardiac cycle). Find the total change in volume from time 0 to time \(T\) (the end of the cardiac cycle). (Your answer will involve the constants \(k, P_{0}, m,\) and \(\left.T .\right)\)

Step-by-Step Solution

Answer

The total change in volume is \(-\frac{k P_{0}}{m} (e^{-m T} - 1)\).

Step 1: Identify the Given Rate Function

The rate of change of blood volume in the aorta is given by the function \(-k P_{0} e^{-m t}\) in milliliters per second, where \(k\), \(P_{0}\), and \(m\) are constants.

Step 2: Set Up the Integral

To find the total change in volume from time 0 to time \(T\), we need to integrate the rate function, \(-k P_{0} e^{-m t}\), over the interval from 0 to \(T\). Thus, we calculate the integral \(\int_{0}^{T} -k P_{0} e^{-m t}\, dt\).

Step 3: Integrate the Function

Integrate the function \(-k P_{0} e^{-m t}\) with respect to \(t\). The integral of \(e^{-m t}\) is \(-\frac{1}{m} e^{-m t}\), so the integral becomes:\[\int_{0}^{T} -k P_{0} e^{-m t}\, dt = \left[-k P_{0} \left(-\frac{1}{m} e^{-m t}\right)\right]_{0}^{T}.\]

Step 4: Evaluate the Integral at the Limits

Substitute the limits of integration into the integrated function:\[\left[-\frac{k P_{0}}{m} e^{-m t}\right]_{0}^{T} = -\frac{k P_{0}}{m} e^{-m T} + \frac{k P_{0}}{m} e^{0}.\]

Step 5: Simplify the Result

Simplify the result to find the total change in volume:\[-\frac{k P_{0}}{m} \left(e^{-m T} - 1\right).\]The total change in volume from time 0 to time \(T\) is thus given by this expression.

Key Concepts

Integral Calculus in Rate of Change

Integral calculus is a powerful mathematical tool used to calculate the total change in a quantity when we know its rate of change. In this exercise, we're looking at how the blood volume in the aorta changes over a cardiac cycle. The rate of change of this volume is given by

• -k P_{0} e^{-m t}

Integrating this expression from time 0 to time \(T\) will give us the total change in volume during one cardiac cycle. Understanding and setting up integrals is foundational when dealing with continuous changing rates, and it allows us to sum these small differences over an interval to find an overall change. In our problem, integrating this function over the limits 0 to \(T\) effectively adds up all the infinitesimal changes in the aorta's volume as time progresses from the start to the end of the cardiac cycle.

Understanding the Cardiac Cycle

The cardiac cycle is a term that refers to the sequence of events that occur from the beginning of one heartbeat to the beginning of the next. It involves the contraction and relaxation of the heart muscles, resulting in the flow of blood.

• Systole: the phase where the heart contracts and pumps blood out.
• Diastole: the phase in which the heart relaxes and fills with blood.

During these phases, specifically within the aorta, blood volume changes due to the elasticity and dynamics of the blood vessels. The exercise reflects these changes by showing how the rate of blood volume change can be modeled mathematically. This rate is affected by factors such as the initial pressure in the aorta and the natural elasticity of the aorta wall, all of which are common characteristics considered in the cardiac cycle.

Calculating Volume Change

Volume change in the context of this problem describes how the amount of blood in the aorta changes over a cardiac cycle. To calculate this change, we use

• the rate of change function \(-k P_{0} e^{-m t}\),

Integrating this function gives us the net change in volume from time 0 to \(T\). The fundamental theorem of calculus tells us an integral of a function represents the accumulation of values of that function over a defined interval. In our example, solving the integral provides a precise expression \[-\frac{k P_{0}}{m} \left(e^{-m T} - 1\right)\]This expression represents how the blood volume fluctuates as it moves through the cardiac cycle, from its initial state to the end of that cycle, based on the constants \(k\), \(P_{0}\), \(m\), and the time \(T\) itself.